CONSTRUCTIONOF STRAIGHT LINES

Book I. Propositions 2 and 3

AFTER STATING THE FIRST PRINCIPLES, we began with the construction of an equilateral triangle. The goal of Euclid's First Book is to prove the remarkable theorem of Pythagoras -- about the squares that are constructed of the sides of a right triangle. We will come to it.

The object of this science is knowledge of figures, and our knowledge consists first in being able to draw the figure. Hence, the Postulates and the early propositions are devoted to what we are able to draw. Yet we draw nothing but straight lines and circles! Therefore, the only figures we can say we know will be those we can draw with straightedge and compass.

We use a straight edge to draw a straight line. A straightedge is not a ruler, it is not used for measurement. It is the compass that becomes the measuring instrument, because with it we can make equal distances.

Use of the straightedge and compass is laid down in the first three Postulates. They are quite explicit -- and quite strict:

1. To draw a straight line from any point to any point.

2. To extend a straight line for as far as we please in a straight line.

3. To draw a circle whose center is the extremity of any straight line,

3. and whose radius is the straight line itself.

Consider now the following problem: At the point A, how may

we draw a straight line equal the straight line BC? How do the postulates permit it? Logically, what are the steps?

You might think that all we do is to adjust the compass to BC and then move it over to A. But there is nothing in the Postulates that allows a compass-carried distance. Postulate 3 means that, to draw a circle, there must already be a straight line to serve as its radius. But at the point A there is no such straight line.

Hence, the duplication of BC at A is a bit challenging.

In practice, of course, we do simply adjust the compass to BC and then move it over to A. We can say, then, that the following proposition logically allows for a compass-carried distance.

Here is the formal proof. As always, to avoid scrolling, the figure will be repeated.

(The construction begins by drawing the equilateral triangle ABD; then extending DA, DB, to E and F; drawing a circle with radius BC, thus making BG equal to BC; then making DG, DL radii of a circle, so that on subtracting DB, DA respectively, AL will equal BG and therefore BC.)

This proposition leads directly to the next one, where we will be required to cut off from the longer of two straight lines a length equal to the shorter line. The solution is obvious -- but notice how we must rely on Proposition 2; line 6 below. In fact, the next proposition is the only one that requires Proposition 2. The "given" point is the endpoint of a line.

The student should now begin to see how each proposition depends on previous propositions. That is the nature of any logical theory. That is the axiomatic method.